with \(h_{ij} = t_{ij} + \sum _{kl} \bar{v}_{ikjl}\rho _{lk}\)  is the mean field
and \(\Delta _{ij} = \frac{1}{2}\sum _{kl} \bar{v}_{ijkl}\kappa _{kl}\)  the pairing
field. Solving (REF ) determines completely the generalized density,
hence \(\rho \)  and \(\kappa \) . Physical observables can be computed by taking the
trace of the corresponding operator with the densities. In the simplest case of
one-body operators \(\hat{F}\) , this means: \(\mathinner {\langle {\hat{F}}\rangle } =\mathrm {Tr} (\hat{F}\rho ) = \sum _{ij} F_{ij}\rho _{ji}\) . The expectation value
of two- and \(N\) -body operators can be computed thanks to the Wick theorem
[1]}, [2]}.
